The hot embossing of thermoplastic polymers has attracted attention as a promising microfabrication process. Hot embossing has certain advantages over other polymer microfabrication processes. The micro-casting of curable liquid resins, which is a process that is used with elastomers such as polydimethylsiloxane (PDMS), is widely known as soft lithography (1) and is ideal for prototyping small numbers of devices. Unfortunately, considerable manual skill is required to handle the highly flexible components produced.
Available techniques for automating soft lithography have so far proved largely elusive. One example of such methods is injection molding. Injection molding may be used to form microscopic features (2) and can easily be automated, but tooling and equipment costs associated with this method are relatively high.
Finite-element numerical modeling of thermoplastic embossing has also received attention in the art. For example, patterning of sub-micrometer-thickness polymeric layers, as encountered in nano-imprint lithography, has been considered (8-14). The embossed material has variously been described using models such as Newtonian liquid (8, 9, 14), shear-thinning liquid (8, 15), linear-elastic (11), Mooney-Rivlin rubber-elastic model (10, 16), and linear (12, 13) or non-linear (17, 18) visco-elastic models. Other thermomechanically-coupled finite-deformation material models have also been developed (19, 20) and applied to simulate the micro-embossing of bulk polymeric substrates (20). However, finite-element approaches, although capable of capturing many of the physical phenomena observed, are currently too computationally costly to extend to the feature-rich patterns of complete devices.
For the simulation of nanoimprint lithography, Zaitsev, et al. (22) have proposed a simplified “coarse-grain” approach in which the imprinted polymeric layer is modeled as a Newtonian fluid and the pattern of the stamp is represented by a matrix of cells, where each is assumed to contain features of a single size and packing density (21-25).
Efficient numerical simulations of the deformations of elastic (26, 27) and elastic-plastic (28-30) bodies, which may be rough and/or multi-layered (26, 29, 30), have also been considered in tribology. These simulations, in the elastic-plastic cases, rely on a description of the deformation of the material's surface in response to a point-load together with a criterion for yielding of the material. The overall topography of the material's surface is calculated by spatially convolving an iteratively-found contact pressure distribution with the point-load response. Sub-surface stresses can similarly be estimated by convolving contact pressures with appropriate kernel functions (27). The convolution itself may be effected using fast Fourier transforms (26, 28, 29) or other summation methods (31, 32). The solution for the contact-pressure distribution may successfully be obtained using iterative conjugate-gradient methods combined with kinematic constraints on the surface deformation (26, 28, 32) or by using methods that seek a minimum of elastic potential energy in the layer (29).
The validity of these contact mechanics-based approaches is limited to cases where surface curvatures remain small and all deflections are a small proportion of any layer's thickness. These linear methods have nevertheless proved to be of great value because of the fast computation that is possible. Lei, et al. suggest using such an approach to represent the micro-embossing of thick, rubbery polymeric layers (33). They develop an approximate analytical expression for the shape of the deformed surface of such a layer when embossed with a simple trench, and show rough agreement between that expression and the measured topography of polymethylmethacrylate layers embossed under a small set of processing conditions.